The answers to most of your questions are under active research and there are many unknowns. Understanding this process is one of the main goals of developmental biology. So, try to set reasonable goals for your project, mainly by seeing where you have sufficient data to work with.
I would start with reading some basic textbook on developmental biology.
Regarding the flow of information: Remember that starting from the zygote and onwards to the whole organism, the DNA sequence is generally the same between all cells. As a computer scientist it might help you to think about it this way:
Think of the DNA as a very complicated mathematical function $f()$ acting on cellular states. The current cellular state $S_i$ will be the total molecular components of the cell and will yield the biological function of the cell. So, you start with an initial cell state $S_0$ (could be a vector, for example) and then the daughter cells will be: $S_1=f(S_0)$, their daughter cells will be $S_2=f(S_1)$. This is not a perfect representation of the process (since at time point 2 you may actually have both $S_0$ and $S_1$ cells), but it emphasizes the point that although the DNA sequence "computes" the next step it is in fact constant, and the point that each cellular state dictates the next state. Seems to me like a Markov chain could be appropriate. Of course you can also try more sophisticated models - but it is better to start simple.
Also, I would suggest starting with C. elegans as your modeled species. This is because the full differentiation fate of each of its 1031 somatic cells is known - a remarkable achievement and something you will not have in other species.
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